The Wealth and Weil-Being Curves, and Political Freedom - A Mathematical Analysis: Part 1 By Patrick Bruskiewich Mathematisches Institut der Vancouver, BC Abstract In a recent newspaper the author came across a scatter plot of Wealth (GDP) versus Well- Being. The distribution is similar to those found in Learning Theory and in a number of Applied Mathematics models. The scatter plot begs a mathematical analysis as well as a correlation to socio-economic measures. 1.0 The Wealth and the Weil-Being Curve In a recent edition of a Canadian newspaper The National Post the author came across an article about the puzzling angst of the People of France. With this March 26, 2013 article was included a distribution that plotted Gross Domestic Product as an independent variable with an implied correlation to a dependent "subjective Well-Being Index" (refer to Fig. 1 : GDP versus Well-Being Index). The distribution shows a number of indicators that relate or correlate to the "subjective Well-Being Index", and representative economic systems. The scatter plot came complete with a curve (denoted red in the article) with no description of the functional form the fit. The distribution was sourced from the Paris School of Economics. A plethora of curves can be fitting to this distribution depending on the variables used. Perhaps, to a first approximation a restricted growth algorithm, or Mitscherlich relationship, could be used to describe the distribution. [1] ■ LATIN AMERICA o FORMER COMMUNIST 4.5 -LB -1.5 -Z.B -2.5 Puerto Ricn o Ireland New Zealand Denmark Iceland .>-. A Switzerland Netherlands • Sweden -a * united States^ Australia -o J^ mximbowg^ Britain BeJoium Finland -o 6 A, o- Austria N°™ay-o Bangladesh lran h India c „ !■ /y o Serbia BurkinaXy' cardan Tanzania ^Azerbaijan ^^ Rwanda o- Estonia Latvia-Q o- Lithuania Georgia - , jv Russia j. Bulgaria -do Alhania-o Romania J Belarus^ ^ Jkraine lraq-©„ O-MoEdcva O- Armenia 9 ZSmbatae Curve with article Fitted SWI Curve ~i ; 1 i 5 10 15 20 GDP PER CAPITA, IN THOUSANDS OF DOLLARS 25 3D SOUH.CK: I'AKIS SCHOOL ("Jl-' ECONOMICS JONATI [ON IlIVAiT / NATIONAL POST Fig. 1: GDP versus Weil-Being Index 2.0 Restrictive Growth and the Mitscherlich Relationship The Mitscherlich Relationship describes a system governed by the differential equation dw dt = k(B-w) where k is a positive constant which determines how fast the differential tends to zero, and where B represents an upper bound. There exists an upper bound B for the productivity of the system (the line y = B is the asymptote). No crop, industry or population can grow or flourish indefinitely. There are limitations set by shortages of raw materials, productivity of land and by some other sort of control mechanism, such as limits to demand, inherent economic efficiencies and the manner in which decisions are made or populations are governed. For example, there is a relationship between fertilizer use and crop yield. To first approximation, the function y can be considered as a monotonically increasing function. If x where the amount of fertilizer applied to the cultivation of a crop, the yield y cannot be raised indefinitely by the application of more and more fertilizer. All things being equal, the land would, with time, become limited in its crop productivity as the soil becomes saturated with fertilizer. For such a system a workable approximation to the output or productivity of the land is given by the Mitscherlich Relationship, y = B(l-e- b ) In analyzing the locus of points that is graphed with the original article, the author was able to fit a simple Mitscherlich Relationship to the red curve which has the form SWI = 7r(\-e kw ) l where k = — , which appears to be a defining feature on the left hand side of the distribution and appears to be an inflection point for the distribution. The plot of the Subjective Well-Being Index (SWI) function is the blue line in Fig. 1. The graph and the SWI fit (blue curve) begs an in-depth mathematical analysis, as well as a correlation on socio-economic indicators. 3.0 Political Freedom as a Significant Socio-Economic Measure Based on a careful reading of history, it is evident that there is a continuum of Political Freedom that spans from Totalitarianism at one extreme, to a full and open Liberal Democratic Institutions and the Ecole Suisse at the other extreme. An index that runs from to 3 can be applied to the Continuum of Political Freedom (refer to Table 1 : Continuum of Political Freedom) 0.5 1.0 1.5 2.0 2.5 3.0 Totalitarian Neo- Communism Autocratic Oligarchy Capitalism Liberal Democracy Ecole Suisse Table 1: Continuum of Political Freedom To more fully represent a correlation between Wealth and Well-Being, while applying a measure of the Continuum of Political Freedom (0 < m < 3), we must look beyond the simplicity of the Mitscherlich Relationship. 4.0 Modeling SWI using the Logistic Equation It would seem reasonable to develop a model that is independent of monetary measures that depend currency exchange are are subject to inflation and other competitive factors. Beyond the Mitscherlich Relationship, restrictive growth can be described by a differential equation of the expanded form where X is a positive coupling constant . The solution for this expanded expression is B y ~\ + ke- m where k has a positive value. For the case k = 1 and e ~ lBt « 1 we get y « B 1 1 - e~ xm j . Based on the SWI functional fit (above) 10 IOtt Using a scaling factor, let t = l0m Then the coupling factor becomes 7t 5.0 Subjective Well-Being and Political Freedom Consider a logistic function of the more complete form dy dm = A'(A-y)(B-y) where m is the Continuum of Political Freedom ( < m < 3 ) outlined above. The solution for this more general expression is the logistic equation y = A + (B-A) l + ke A'(B-A)m It is worth noting that n appears as the factor B in the Mitscherlich Relationship. Our more developed model should involve universal measures such a n . We can do this by choosing a function of the form for the Subjective Well-Being Index, SWI = -2.0 + 2n \ + ke -2.0m Such a function should describe a span from a well-being index of -1.5 to 4.25 from whence we conjecture that the factor k is of the form k = 7u { *- x) * 11.606 The inflection point is at d r ^,\ dm dy ]_ d ro , V dm ) dm \A'(A-y)(B-y)] = which occurs at the value y = \(B + A) which yields y = (7U-2) = \. 14159 For SWI as a function of Political Freedom ( < m < 3 ) we find the following (refer to Table 2: SWI as a function of the Political Freedom m): Continuum Measure m Political Freedom Subjective Well- Being Index Efficiency Totalitarian -1.5 -0.37 0.5 Neo- Communism -0.81 -0.20 1.0 Autocratic 0.44 0.11 1.5 Oligarchy 1.98 0.48 2.0 Capitalism 3.18 0.78 2.5 Liberal Democracy 3.82 0.93 3.0 Ecole Suisse 4.10 1.0 Table 2: SWI as a function of the Political Freedom Measure m A graph of the Subjective Well-Being Index (SWI) as a function of the Political Freedom Measure m is given in Fig. 2: The SWI as a function of the Political Freedom Measure m with some representative countries plotted on the distribution. It is clearly evident the more open forms of government not only provide for a better sense of Well-Being and Quality of Life they also, by the nature of the Logistic equation, provide for greater opportunity for growth and for economic efficiency. SWI as a function of m -' Denmark Suisse Ideal Brazil ^* -*S nl j, * USA Taiwan _*_ •» "OTC Singapore wf Tig Spain W Japan China* Poland ^J" South Korea South Africa ♦ India W Hungary 5/ Russia ' : ■JJ 1 Ukraine •ff North Korea m (Political Frasflom) 35 1 Fig. 2: The SWI as a function of the Political Freedom Measure m A measure of efficiency is the ratio of the SWI for a Political Freedom Measure m compared to the Ideal Measure at m = 4.0 (refer to Fig. 3: Economic Efficiency as a Function of the Political Freedom Measure m). Clearly the Totalitarian and the neo-Communism Regime are less than efficient in their provisioning of goods and services compared to the Open Capitalism, the countries with Parliamentary and Liberal Democratic ideals, and the Ecole Suisse. 6.0 Learning, Training and Development In a real sense the SWI function describes the base and the enlightened extremes of human behaviour. One could speculate that the Continuum Measure m is a measure of Informed -Consent, and a measure s which represents an Efficiency of Governance. Bf iciency as a f Lncii on of m 120 1.00 30 60 fc 040 020 ^ 000 -3 20 -: io -0 60 m {Ported fT««dom|i $ulm IJmI I CW, ^*^^"*1HA s / ^L South *V#» > 1 1 aiy^ 1 1,5 2 2 5 J 5 5 : *m»» WeithKfrru Fig. 3: Economic Efficiency as a Function of the Political Freedom Measure m For instance, for a measure of 1 < / < 3 and 1 < C < 3 , a simple model for m would be 9 V ; and where < s < 1 . The form of the SWI Function is similar to mathematical functions that describe Learning, Training and Development. This may be touched upon in a subsequent paper. You will note the anomalous nature of Communist North Korea under Kim and Neo- Communist Russia under Putin, compared to for instance the People's Republic of China. References: [1] E. Batschelet, Introduction to Mathematics for Life Scientists, Springer- Verlag,